An elementary example of Sard's Theorem sharpness

TL;DR

This paper presents a simple example of a $C^1$ function demonstrating that Sard's Theorem's differentiability condition cannot be lowered, highlighting the theorem's sharpness with an accessible illustration.

Contribution

It provides an elementary, explicit example of a $C^1$ function with a critical value set of positive measure, illustrating the sharpness of Sard's Theorem.

Findings

The constructed function has a critical value set of positive measure.

The example is simpler than typical constructions in the literature.

It confirms the optimal differentiability condition in Sard's Theorem.

Abstract

In this note we define a $C^1$ function $F:[0,M]^2\to [0,2]$ that satisfies that its set of critical values has positive measure. This function provides an example, easier than those that usually appear in the literature, of how the order of differentiability required in Sard's Theorem cannot be improved